Gray $\mathcal{A}$-metrics on nilpotent Lie groups and the quasicentroid condition

Abstract

We study left-invariant Gray $\mathcal{A}$-metrics (cyclic-parallel Ricci tensor) on nilpotent Lie groups, focusing on the algebraic condition that the Ricci operator belongs to the quasicentroid $\operatorname{QC}(\mathfrak{n})$ of the Lie algebra $\mathfrak{n}$. We review and develop the abstract structure theory of quasicentroids, and prove a Central Decomposition Theorem: any nilpotent Riemannian Lie algebra with $\operatorname{Ric} \in \operatorname{QC}(\mathfrak{n})$ decomposes as a central product of ideals $\mathfrak{i}_j = E_{λ_j} \oplus \mathfrak{z}$, where $\mathfrak{z}$ is the center of $\mathfrak{n}$ and $E_{λ_j}$ are the negative Ricci eigenspaces. Endowed with the restricted metric, each factor $\mathfrak{i}_j$ is itself a Gray $\mathcal{A}$-space. Finally, we investigate when the general $\mathcal{A}$-metric condition reduces to $\operatorname{Ric} \in \operatorname{QC}(\mathfrak{n})$. We prove this equivalence unconditionally for two-step nilpotent algebras and whenever the metric admits a nice basis, and provide a general characterization via the vanishing of two trace identities.